What is the difference between "either A is true or A is false" and "either A is true or ~A is true?" I have an intuitive sense that they are two very different statements but I am having a hard time putting why they are different into words. Thank you.

I think you're getting at the difference between the principle of Bivalence (there are only two truth values—true and false) and the Law of Excluded Middle: 'P or not-P' is always true.

Suppose there are some sentences that are neither true nor false. That might be because they are vague, for example. It might not be true to say that Smith is bald, but it might not be false either; it might be indeterminate. So if S stands for "Smith is bald," then "Either S is true or S is false" would not be correct. Our assumption is that S isn't true, but also isn't false. However, if by "not-S" we mean "S isn't true," then "S or not-S" is true. That is, bivalence would fail, but excluded middle wouldn't.

But as you might imagine, there's a good deal of argument about the right thing to say here.

I presume that you're using the formula "~ A" to abbreviate "It is not true that A" rather than "It is false that A." If my presumption is wrong, then this response may not answer your question.

Where A is some proposition, I see no difference between "It is not true that A" and "It is false that A": Every proposition that isn't true is false, and every proposition that isn't false is true.

However, the same doesn't hold if A is, instead, some sentence. For a sentence can fail to be true without being false. To use an admittedly controversial example: the self-referential sentence "This sentence is not true" is neither true nor false, because the sentence fails to express any proposition in the first place (including the proposition that the sentence isn't true!). Any false sentence is not true, but a sentence can fail to be true without being false.

But perhaps you meant to use the formula "~ A" to represent rejection or denial of the sentence or proposition A. Some philosophers distinguish between (1) rejecting or denying A, on the one hand, and (2) asserting (something) that (implies that) A is false, on the other. See this SEP entry. I myself don't find their view plausible.

Perhaps I could add something here too—and perhaps it will be useful: You are right that there is a difference between the two statements that you offer, and the difference has become more significant with the rise of many-valued logics in the 20th and 21st centuries.

If one says, “A is either true or false,” then there are only two possible values that A can have—true or false. But if one says, “either A or not-A is true,” then there might be all sorts of values that A could have: true, false, indeterminate, probably true, slightly true, kind of true, true in Euclidean but not Riemannian geometry, and so on. The first formulation allows only one alternative to “true” (namely, “false”), but the second formulation allows many alternatives. The second formulation does indeed require that at least A or not-A be true, but it puts no further restrictions on what other values might substitute for “true.” (For example, perhaps A is true, and yet not-A is merely indeterminate.)

The advantage of sticking to the first formulation (often called the principle of bivalence) is that it forces us to reason from propositions that describe what is definitely so or not so, and as a result, we can actually prove things. (After all, if we were to give reasons that were neither true nor false, then our reasons would seem to end up proving nothing. Imagine, for example, someone saying, “I believe this conclusion for a good reason, but my reason is neither true nor false.” Moreover, if the conclusions we wanted to prove were also to turn out to be neither true nor false, then they would remain unprovable; what would it mean, one might ask, to “prove” the untrue?) Considerations of this sort led Aristotle to believe that scientific knowledge always depended crucially on propositions of argument that had to be true or false.

On the other hand, there are many situations in life where our ideas are so vague and indefinite that the best we can say is that a particular proposition seems somewhat true, or true to a certain degree, or true for the most part. (For example, Aristotle held that propositions of ethics were sometimes only “true for the most part.” In the Middle Ages, a number of logicians wanted to use “indeterminate” as a truth value, in addition to true and false, and in the 20th and 21st centuries, logicians have experimented increasingly with the idea that there could be many truth values, in addition to true and false. As a result, there are now various systems of many-valued logic, including so-called fuzzy logic, which assigns numerical degrees of truth to different propositions.)

All the same, the principle of bivalence still plays a fundamental role even in systems of many-valued logic, albeit at a higher level. (The second formulation that you have cited is now termed the law of excluded middle, though before the development of many-valued logics, the two formulations would have amounted to the same thing.)

Specifically, many-valued logics assign different values to various propositions and then draw conclusions from the assignments. (For example, if A is “somewhat true,” then one can conclude that A is not “entirely false.”) Nevertheless, such systems always rely on at least two crucial assumptions: (1) the propositions in question, such as A, do indeed have the assigned values or they do not, and (2) these propositions cannot both have the assigned values and not have them. The first assumption is the principle of bivalence all over again, though at the “meta” level (meaning that it applies, not to A, but to statements about A, that is, to the statements of A’s truth value). And the second assumption is the traditional law of contradiction. (For more on the law of contradiction, you might see Questions 5536 and 5777.)

In other words, the propositions treated by a system of many-valued logic are typically imprecise and indefinite, and what a many-valued logic then does is allow us to talk in a precise and definite way about the imprecise and indefinite. To achieve this result, however, the system’s own statements must be definite, and to achieve coherence, the system’s own statements must also be noncontradictory. By contrast, if one were to relax these restrictions on the system, then all one would get would be an indefinite discussion of the indefinite, or an incoherent discussion. And if this last result were all that one hoped to achieve, then there would be no need to build the system in the first place. Instead, just leap from bed in the morning, and without drinking any tea or coffee, start talking. If you are like me, you will then arrive almost instantly at the appropriate level of grogginess.